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How Astronomers Cracked the Einstein Code

Adam Frank. Astronomy. Volume 36, Issue 9. September 2008.

Simulating the merger of black holes is no small scientific feat. For one thing, it requires a lot of computing horsepower the kind you can get only from a supercomputer. It also involves solving the complex, multilayered equations of Einslein’s theory of general relativity at lightning speed. Using supercomputers, researchers in the field of numerical relativity hope to shed light on the physics of the most violent events in the cosmos, such as black-hole mergers and colliding neutron stars.

But numerical relativity demands that researchers take on Albert Einstein full bore. Solving the equations of general relativity—Einstein’s grand theory of gravity—is difficult enough. The additional challenge is to digitally animate the most complex behaviors hiding inside those equations.

Getting results has become urgent. A new kind of telescope—gravitational wave detectors, built to sense ripples in the fabric of space and time—has recently come on line. These gravity observatories were built specifically to detect events like black-hole mergers. But the telescopes need guidance from computer models to know what sort of signal to look for. This special imperative has made numerical relativity the kind of challenge that draws theorists with a special kind of ambition.

For years, the computer codes at the heart of numerical relativity simulations could do little more than simply crash. All paths seemed blocked; nothing worked. In spite of all the intellectual firepower attracted to the research, and 30 years of work, some theorists were giving up hope. Simulating Einstein’s universe might not just be hard—it might be impossible.

But recent breakthroughs in computer modeling by numerical relativists mean the simulations are finally working. They found a way to run Einstein’s equations as black holes collide, without crashing their supercomputers. Animations created from the model runs show black holes spiraling into each other and merging while emitting blasts of radiation and rippling gravity waves.

This research has opened a new window on the universe—one in which astronomers can hopefully observe disturbances in space-time as readily as flashes of X rays and visible light.

Listening for Black Holes

On the dusty plains of the Hanford Nuclear Reservation, about 500 miles (800 kilometers) southeast of Seattle stands numerical relativity’s reason for existence. It consists of two long vacuum chambers that meet to form an L-shaped gravity observatory. The facility is called LIGO, the Laser Interferometer Gravitational-wave Observatory.

The LIGO Hanford Observatory, along with a sister facility, the LIGO Livingston Observatory in Louisiana, represent a new kind of telescope.

LIGO is designed to pick up gravitational waves—ripples on the surface of space-time’s ocean that are a key prediction of Einstein’s general theory of relativity. Wiggling a mass back and forth produces gravity waves that move across the fabric of space-time. Wiggling a massive, compact object like a black hole creates gravity waves so powerful they can be detected across astronomical distances.

“Gravity waves are the point of LlGO,” says Scott A. Hughes, a professor of physics at MIT’s Kavli Institute for Astrophysics and Space Research. “The crossed lasers are designed to detect changes in space-time produced by passing gravity waves. But you have to know what to look for.”

When people started thinking about LIGO, they realized that the merger of two orbiting black holes would produce a strong signal, A binary system of two massive stars orbiting each other sets the stage for the merger. When the stars go supernova, they each leave behind a black-hole corpse.

The black holes remain locked in orbit around each other. The closer they get, the more they disturb local space-time. The system generates gravity waves that ripple outward. The waves sap orbital energy from the black holes, which then circle each other ever more closely. They eventually merge into a single object in a cataclysm of distorted space-time and radiating gravity waves.

“LIGO should be able to see these mergers,” Hughes explains. The LlGO project set researchers on the black-hole merger problem with a vengeance. For years, scientists have been trying to use general relativity to calculate the signals LIGO scientists should look for in their data. A match would mean LIGO has detected a black-hole merger.

Death Spiral

There are three phases to a black-hole merger. First, “inspirai” occurs as the black holes circle around each other on an ever-tightening orbit. Until they get close, slight modifications of Newtonian gravitational attraction work just fine.

At the middle phase of the merger, all hell breaks loose. Tracking two black holes as they spiral into collision demands the full-blown equations of general relativity with no tricks or simplifications. The merger also produces the strongest LIGO signal. The melding of two rapidly moving black holes shreds space-time and radiates a torrent of gravity waves from the scene.

Like the inspirai phase, the merger’s conclusion is relatively easy to calculate. “You end up with a single black hole,” says Hughes. “The event horizon of the merged black hole oscillates for awhile and radiates gravity waves until it stabilizes.” This final phase of black-hole merger is called the ringdown.

Accurately reproducing the details of a merger and its gravity wave signature is the grand challenge needed to satisfy LIGO. Earlier this decade, a merger was the prize everyone was gunning for. Winning that prize meant Einstein’s equations would have to be wrestled to the mat.

Einstein’s Impossible Legacy

It’s the complexity of relativity theory that makes numerical relativity so hard. To handle a problem like merging black holes, theorists have to solve 10 interwoven equations simultaneously. Take just a few steps into a calculation and you can end up with hundreds of terms smaller pieces of equations—to follow. It’s like doing algebra and calculus in the middle of a tornado.

Theorists had to first think of ways to convert the equations of general relativity into a form that computers could swallow. “The first attempt at numerical relativity came in the 1970s,” says Joan M. Centrella, chief of the gravitational astrophysics laboratory at Goddard Space Flight Center in Greenbelt, Maryland. “It was really a heroic effort.”

Centrella understands such theoretical heroism. She and her research group have worked on numerical relativity simulations for 20 years. The first efforts focused on direct collisions of black holes. The calculations were crude, but they did point the way for a future generation of researchers. To make real progress, scientists would have to figure out how to simulate Einstein’s 4-D spacetime on a computer and make it work for the most complicated situations.

In general relativity, space and time are intrinsically mixed. All objects in space-time, including you and me, have four dimensions. Each of us occupies and moves through the three dimensions of space and, simultaneously, a fourth dimension of time. That means each person’s life history is a four-dimensional object stretching from birth to death. The same principle applies to black holes.

Black Hole Weirdness

To make it possible to run the equations of general relativity in a computer, researchers had to develop a way to accommodate the complexities of four-dimensional objects. Then, they had to develop a way to visualize the data as 2-D and 3-D animations. This left another major challenge: simulating the black holes themselves.

Every black hole is surrounded by an event horizon, the point beyond which ordinary radiation cannot escape the objects powerful gravity. The event horizon is the boundary between our universe and the weirdness inside. Anything passing through the horizon exits our world forever. It’s no surprise, then, that putting black holes into computer simulations creates problems.

Inside of the event horizon is a place where simulations can’t go. Yet, without a way of representing black holes in the model, there could be no numerical relativity. Over the years two strategies emerged to treat the problem. “You can either cut the black holes out of the grid in a process called excising, or you can try and slow the evolution down near the hole and then insert a solution you already know, which is called puncture,” explains physicist Manuela Campanelli, director of the Center for Computational Relativity and Gravitation at the Rochester Institute of Technology (RIT) in New York.

Each choice has its problems, and neither worked well. The difficulties of computational relativity brought the field to a standstill. Only a few years ago, the situation appeared to be getting desperate. “The codes just crashed and crashed and crashed,” Centrella says. “You couldn’t even get an entire orbit out of them.”

The inherent mathematical nastiness of general relativity’s equations and the sheer difficulty of translating them into computer code rendered the models wildly unstable. Just a few steps into a calculation, the machines locked up by trying to divide by zero or some other impossibility.

Campanelli remembers it as a dark time. “People had really lost hope,” she says. “These were scientists who had spent years building their codes. All that work, all that mathematics. No one wanted to toss it away and start again.”

Then, in the midst of the melancholy, everything changed. It was a day that numerical relativists are not likely to forget for a long time.

The Lone Gunman

On April 19, 2005, numerical relativists gathered at a conference center in the mountains of Banff, Alberta. Frans Pretorius, a professor of physics at Princeton University, showed up with a secret. “Frans was the classic lone gunman,” Hughes says. “He appeared at a conference and said ‘Watch this.'”

After describing the mathematical background of his numerical code, Pretorius showed the audience a simulation with five full orbits of a black-hole pair. “It was just amazing,” Hughes says. “Frans had solved it. He got it to work. It was like everybody was climbing this mountain only to find Frans on top already waving down at them.”

In Pretorius’ new method, Einstein’s equations take on a form similar to those describing simple waves. It was a highly abstract, nonintuitive approach. But progress was not immediate. “My first attempt failed,” he recalls. “In the second attempt, things were a little better.”

News of Pretorius’ achievement had leaked out before his talk. The buzz had already started when he arrived at the conference hall. “People were text messaging during my talk,” he recalls. “Some guy set up with a video camera on a tripod and aimed it right at me. It was kind of embarrassing.”

After Pretorius showed his animations and finished his talk, the community tried to digest what it had just seen. A few people were hostile. “There was a lot of investment in the methods researchers had been working with,” Pretorius says. “Not everyone was pleased that I had come at it in an entirely different way and succeeded.”

But most of Pretorius colleagues were thrilled. He had proved that numerical relativity was not an impossible dream. What happened next was more than anyone could have expected.

Changing Course

In the wake of Pretorius’ success, all of the researchers asked themselves the same question. “If Frans got it to work his way,” Centrella explains, “then did we all need to change to Frans’ method?” Some researchers decided to press ahead with their existing codes.

The RIT group members decided to focus efforts on their puncture method for treating moving black holes. It had been a major source of problems for them, but they came up with a solution.

Many in the field of numerical relativity believed that a puncture representing a black hole could not move. So researchers would just pin the puncture in the grid and let space-time move around it.

Campanelli and her colleagues decided to allow the puncture to move. To everyone’s surprise, the method produced spectacular results. Suddenly, Campanelli’s group could track orbiting black holes all the way to merger.

The group presented the new success at the next numerical relativity meeting, but met with disbelief. “People saw our simulations but didn’t understand how you could move the black hole across the grid,” Campanelli says.

It might have gone worse for Campanelli s group if Joan Centrella’s team hadn’t shown up with simulations that used the same method. “The community was just stunned silent,” Centrella recalls.

The simulation of black-hole mergers, from inspiral to ringdown, are now routine. “Now there are sessions at the American Astronomical Society meetings dedicated entirely to mergers,” Centrella says. “And just a little while ago, we could not even get an orbit.”

Black Holes, Bright Future

Researchers are moving quickly to explore additional black-hole merger scenarios, including mergers of black holes of different size and spin. “It’s a feeding frenzy right now,” Pretorius says. “And it will still take a few years until all the important cases have been mapped out.”

Already, new surprises have appeared. Research with computer models shows that after a merger, the new, combined black hole experiences a powerful recoil force. In simulations, black holes blast away after mergers at speeds topping 625 miles (1,000 km) per second.

Simulated black-hole mergers can now provide guidance to LIGO and other gravity-wave observatories around the world. Sometime around 2013, LIGO will be upgraded to a more sensitive design, called Advanced LIGO. The improvements in sensitivity should finally give astronomers a telescope ready to detect black-hole mergers.

It has been a long, hard climb for LIGO Observatory scientists as they ready their detectors to hear passing gravitational waves. Numerical relativists experienced their own hard climb, and they are finally getting some satisfaction. They will be ready and waiting as LIGO turns its ear to the sky to detect the faint ripples of distant cataclysms.